One mass falls around another and traces an ellipse — not by decree but by three unbreakable laws. The engine integrates the two-body problem under inverse-square gravity with a symplectic map, then reads the conserved quantities straight off the trajectory: the conic, the swept areas, the period ratio. Rendered, not quoted.
source Kepler, Astronomia Nova (1609) · Harmonices Mundi (1619) — archive.org/details/astronomianovaai00kepl (facsimile; stable IA mirror)
A point mass at position r moves under Newton's inverse-square pull toward a fixed centre (reduced two-body form, GM = 1):
Two quantities never change along the path — the specific energy and the angular momentum:
From them alone the shape follows: a conic with the centre at a focus, semi-major axis a = −GM/2E and eccentricity e = √(1 + 2EL²/GM²).
The closed ellipse is the two-body case of what the-n-body traces. Take Newton's gravity, keep two masses, and Kepler's three empirical laws fall out as theorems — conservation turned into geometry.
Add a third mass and the ellipse stops closing: it precesses, it wanders. Kepler's exact ellipse is the reachable island in the n-body sea — the case that has closed-form conserved quantities.
Live re-check: integrate one radial period and measure the apsidal angle Δφ swept perihelion-to-perihelion. A genuine inverse-square orbit closes at exactly 2π. If window 6 tampers the force law, the orbit precesses and this flips red.
Initial state at perihelion, GM = 1, symplectic step dt = 0.0012:
Gold sectors are swept in equal times — equal areas (Kepler II). The star sits at a focus, not the centre.
Proven off the trajectory, not assumed:
True, and honest: this is the two-body Newtonian idealization. The residual is real physics the model omits —
Inside its stated frame the ellipse is exact — and the checks below prove it.
Orbits are perfect circles on nested spheres. → ellipses; the circle is the e=0 special case (Kepler I).
A planet moves at constant speed along its path. → it sweeps equal areas in equal times — fast at perihelion, slow at aphelion (Kepler II).
The Sun sits at the centre of the orbit. → at a focus; the other focus is empty.
Period depends on shape or eccentricity. → T²/a³ is one constant for every orbit, whatever its e (Kepler III).
The disclosed planted void: swap the force law to inverse-r (1/r instead of 1/r²). By Bertrand's theorem only 1/r² and Hooke's law give closed orbits — so the ellipse should precess and Kepler III should break. The WITNESS in window 7 catches it live.